Hostname: page-component-89b8bd64d-sd5qd Total loading time: 0 Render date: 2026-05-06T06:19:54.130Z Has data issue: false hasContentIssue false
  • English
  • Français

The laminar generalized Stokes layer and turbulent drag reduction

Published online by Cambridge University Press:  16 November 2010

MAURIZIO QUADRIO  and
PIERRE RICCO
MAURIZIO QUADRIO*
Affiliation:
Dipartimento di Ingegneria Aerospaziale, Politecnico di Milano, via La Masa 34, 20156 Milano, Italy
PIERRE RICCO
Affiliation:
Department of Mechanical Engineering, King's College London, Strand, London WC2R 2LS, UK
*
†Email address for correspondence: maurizio.quadrio@polimi.it
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper considers plane channel flow modified by waves of spanwise velocity applied at the wall and travelling along the streamwise direction. Both laminar and turbulent regimes for the streamwise flow are studied. When the streamwise flow is laminar, it is unaffected by the spanwise flow induced by the waves. This flow is a thin, unsteady and streamwise-modulated boundary layer that can be expressed in terms of the Airy function of the first kind. We name it the generalized Stokes layer because it reduces to the classical oscillating Stokes layer in the limit of infinite wave speed. When the streamwise flow is turbulent, the laminar generalized Stokes layer solution describes well the space-averaged turbulent spanwise flow, provided that the phase speed of the waves is sufficiently different from the turbulent convection velocity, and that the time scale of the forcing is smaller than the life time of the near-wall turbulent structures. Under these conditions, the drag reduction is found to scale with the Stokes layer thickness, which renders the laminar solution instrumental for the analysis of the turbulent flow. A classification of the turbulent flow regimes induced by the waves is presented by comparing parameters related to the forcing conditions with the space and time scales of the turbulent flow.

JFM classification

Flow Control: Drag reduction Turbulent Flows: Turbulence control

Information

Type
Papers
Information
Journal of Fluid Mechanics , Volume 667 , 25 January 2011 , pp. 135 - 157
Copyright
Copyright © Cambridge University Press 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

REFERENCES

Abramowitz, M. & Stegun, I. A. 1964 Handbook of Mathematical Functions. Applied Mathematics Series 55. National Bureau of Standards.Google Scholar
Baron, A. & Quadrio, M. 1996 Turbulent drag reduction by spanwise wall oscillations. Appl. Sci. Res. 55, 311–326.CrossRefGoogle Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Choi, J.-I., Xu, C.-X. & Sung, H. J. 2002 Drag reduction by spanwise wall oscillation in wall-bounded turbulent flows. AIAA J. 40 (5), 842–850.CrossRefGoogle Scholar
Choi, K.-S. 2002 Near-wall structure of turbulent boundary layer with spanwise-wall oscillation. Phys. Fluids 14 (7), 2530–2542.CrossRefGoogle Scholar
Coleman, G. N., Kim, J. & Le, A. T. 1996 A numerical study of three-dimensional wall-bounded flows. Intl J. Heat Fluid Flow 17, 333–342.CrossRefGoogle Scholar
Dean, R. B. 1978 Reynolds number dependence of skin friction and other bulk flow variables in two-dimensional rectangular duct flow. Trans. ASME I: J. Fluids Engng 100, 215.Google Scholar
Du, Y. & Karniadakis, G. E. 2000 Suppressing wall turbulence by means of a transverse traveling wave. Science 288, 1230–1234.CrossRefGoogle ScholarPubMed
Du, Y., Symeonidis, V. & Karniadakis, G. E. 2002 Drag reduction in wall-bounded turbulence via a transverse travelling wave. J. Fluid Mech. 457, 1–34.CrossRefGoogle Scholar
Dunham, W. 1990 Cardano and the solution of the cubic equation. In Journey through Genius: The Great Theorems of Mathematics, Chap. 6. Wiley.Google Scholar
Fukagata, K., Iwamoto, K. & Kasagi, N. 2002 Contribution of Reynolds stress distribution to the skin friction in wall-bounded flows. Phys. Fluids 14 (11), L73–L76.CrossRefGoogle Scholar
Goldstein, S. 1930 Concerning some solutions of the boundary layer equations in hydrodynamics. Proc. Camb. Phil. Soc. 26, 1–30.CrossRefGoogle Scholar
Itoh, M., Tamano, S., Yokota, K. & Taniguchi, S. 2006 Drag reduction in a turbulent boundary layer on a flexible sheet undergoing a spanwise traveling wave motion. J. Turbulence 7 (27), 1–17.CrossRefGoogle Scholar
Jung, W. J., Mangiavacchi, N. & Akhavan, R. 1992 Suppression of turbulence in wall-bounded flows by high-frequency spanwise oscillations. Phys. Fluids A 4 (8), 1605–1607.CrossRefGoogle Scholar
Karniadakis, G. E. & Choi, K.-S. 2003 Mechanisms on transverse motions in turbulent wall flows. Ann. Rev. Fluid Mech. 35, 45–62.CrossRefGoogle Scholar
Kasagi, N., Suzuki, Y. & Fukagata, K. 2009 Microelectromechanical systems-based feedback control of turbulence for drag reduction. Annu. Rev. Fluid Mech. 41, 231–251.Google Scholar
Kim, J. & Hussain, F. 1993 Propagation velocity of perturbations in turbulent channel flow. Phys. Fluids A 5 (3), 695–706.CrossRefGoogle Scholar
Laadhari, F., Skandaji, L. & Morel, R. 1994 Turbulence reduction in a boundary layer by a local spanwise oscillating surface. Phys. Fluids 6 (10), 3218–3220.CrossRefGoogle Scholar
Marcus, P. S. 1977 On Green's functions for small disturbances of plane Couette flow. J. Fluid Mech. 79 (3), 525–534.CrossRefGoogle Scholar
Marusic, I., Joseph, D. D. & Mahesh, K. 2007 Laminar and turbulent comparisons for channel flow and flow control. J. Fluid Mech. 570, 467–477.Google Scholar
Orr, W. M. F. 1907 The stability or instability of the steady motions of a perfect liquid and of a viscous liquid. Part II. A viscous liquid. Proc. R. Irish Acad. Sect. A: Math. Phys. Sci. 27, 69–138.Google Scholar
Quadrio, M., Auteri, F., Baron, A., Belan, M. & Bertolucci, A. 2009 a Experimental assessment of turbulent drag reduction by wall traveling waves. In Advances in Turbulence XII, Proceedings of the 12th EUROMECH European Turbulence Conference (ed. Eckhardt, B.), vol. 132. Springer.Google Scholar
Quadrio, M., Ricco, P. & Viotti, C. 2009 b Streamwise-traveling waves of spanwise wall velocity for turbulent drag reduction. J. Fluid Mech. 627, 161–178.CrossRefGoogle Scholar
Quadrio, M. & Ricco, P. 2003 Initial response of a turbulent channel flow to spanwise oscillation of the walls. J. Turbulence 4 (7), 1–23.CrossRefGoogle Scholar
Quadrio, M. & Ricco, P. 2004 Critical assessment of turbulent drag reduction through spanwise wall oscillation. J. Fluid Mech. 521, 251–271.CrossRefGoogle Scholar
Quadrio, M. & Sibilla, S. 2000 Numerical simulation of turbulent flow in a pipe oscillating around its axis. J. Fluid Mech. 424, 217–241.CrossRefGoogle Scholar
Ricco, P. & Quadrio, M. 2008 Wall-oscillation conditions for drag reduction in turbulent channel flow. Intl J. Heat Fluid Flow 29, 601–612.Google Scholar
Ricco, P. & Wu, S. 2004 On the effects of lateral wall oscillations on a turbulent boundary layer. Exp. Therm. Fluid Sci. 29 (1), 41–52.Google Scholar
Ting, L. 1960 Boundary layer over a flat plate in presence of shear flow. Phys. Fluids 3 (1), 78–81.CrossRefGoogle Scholar
Viotti, C., Quadrio, M. & Luchini, P. 2009 Streamwise oscillation of spanwise velocity at the wall of a channel for turbulent drag reduction. Phys. Fluids 21, 115109.CrossRefGoogle Scholar
Xu, C.-X. & Huang, W.-X. 2005 Transient response of Reynolds stress transport to spanwise wall oscillation in a turbulent channel flow. Phys. Fluids 17, 018101.CrossRefGoogle Scholar
Yoshino, T., Suzuki, Y. & Kasagi, N. 2008 Feedback control of turbulence air channel flow with distributed micro-sensors and actuators. J. Fluid Sci. Technol. 3, 137–148.CrossRefGoogle Scholar
Zhao, H., Wu, J.-Z. & Luo, J.-S. 2004 Turbulent drag reduction by traveling wave of flexible wall. Fluid Dyn. Res. 34, 175–198.CrossRefGoogle Scholar